Divide the following complex numbers. $\dfrac{30+20i}{1+5i}$
We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate, which is ${1-5i}$. $ \dfrac{30+20i}{1+5i} = \dfrac{30+20i}{1+5i} \cdot \dfrac{{1-5i}}{{1-5i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$. $ = \dfrac{(30+20i) \cdot (1-5i)} {1^2 - (5i)^2} $ Evaluate the squares in the denominator and subtract them. $ = \dfrac{(30+20i) \cdot (1-5i)} {(1)^2 - (5i)^2} $ $ = \dfrac{(30+20i) \cdot (1-5i)} {1 + 25} $ $ = \dfrac{(30+20i) \cdot (1-5i)} {26} $ The denominator now doesn't contain any imaginary unit multiples, so it is a real number. Note that when a complex number, $a + bi$ is multiplied by its conjugate, the product is always $a^2 + b^2$. Now, we can multiply out the two factors in the numerator. $ \dfrac{({30+20i}) \cdot ({1-5i})} {26} $ $ = \dfrac{{30} \cdot {1} + {20} \cdot {1 i} + {30} \cdot {-5 i} + {20} \cdot {-5 i^2}} {26} $ $ = \dfrac{30 + 20i - 150i - 100 i^2} {26} $ Finally, simplify the fraction. $ \dfrac{30 + 20i - 150i + 100} {26} = \dfrac{130 - 130i} {26} = 5-5i $